Vol. 2, No. 3, 2009

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

Terence Tao

Vol. 2 (2009), No. 3, 361–366
Abstract

Let d 3. We consider the global Cauchy problem for the generalized Navier–Stokes system

tu + (u )u = D2u p, u = 0,u(0,x) = u 0(x)

for u : + × d d and p : + × d , where u0 : d d is smooth and divergence free, and D is a Fourier multiplier whose symbol m : d + is nonnegative; the case m(ξ) = |ξ| is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes m(ξ) = |ξ|α for α (d + 2)4. We improve this slightly by establishing global regularity under the slightly weaker condition that m(ξ) |ξ|(d+2)4g(|ξ|) for all sufficiently large ξ and some nondecreasing function g : + + such that 1ds(sg(s)4) = +. In particular, the results apply for the logarithmically supercritical dissipation m(ξ) := |ξ|(d+2)4log(2 + |ξ|2)14.

Keywords
Navier–Stokes, energy method
Mathematical Subject Classification 2000
Primary: 35Q30
Milestones
Received: 16 June 2009
Revised: 22 September 2009
Accepted: 23 October 2009
Published: 9 February 2010
Authors
Terence Tao
University of California
Department of Mathematics
Los Angeles, CA 90095-1555
United States
http://www.math.ucla.edu/~tao/